Ordinary differential equation

ODE redirects here. For the real-time physics engine, see open dynamics engine.

In mathematics, and particularly in analysis, an ordinary differential equation (or ODE) is an equation that involves the derivatives of an unknown function of one variable. A simple example of an ordinary differential equation is

$f'=f\,$,

where $f\,$ is an unknown function, and $f'\,$ is its derivative.

See differential calculus and integral calculus for basic calculus background.

Definition

Let y represent an unknown function of x, and let

$y',y'',\ \dots ,\ y^{(n)}$

denote the derivatives

${\frac {dy}{dx}},\ {\frac {d^{2}y}{dx^{2}}},\ \dots ,\ {\frac {d^{n}y}{dx^{n}}}.$

An ordinary differential equation (ODE) is an equation involving

$x,\ y,\ y',\ y'',\ \dots$.

The order of a differential equation is the order $n$ of the highest derivative that appears.

A solution of an ODE is a function y(x) whose derivatives satisfy the equation. Such a function is not guaranteed to exist and, if it does exist, is usually not unique.

When a differential equation of order n has the form

$F\left(x,y',y'',\ \dots ,\ y^{(n)}\right)=0$

it is called an implicit differential equation whereas the form

$F\left(x,y',y'',\ \dots ,\ y^{(n-1)}\right)=y^{(n)}$

is called an explicit differential equation.

A differential equation not depending on x is called autonomous, and one with no terms depending only on x is called homogeneous.

General application

An important special case is when the equations do not involve $x$. These differential equations may be represented as vector fields. This type of differential equations has the property that space can be divided into equivalence classes based on whether two points lie on the same solution curve. Since the laws of physics are believed not to change with time, the physical world is governed by such differential equations. (See also symplectic topology for abstract discussion.)

The problem of solving a differential equation is to find the function $y$ whose derivatives satisfy the equation. For example, the differential equation

$y''+y=0\,$

has the general solution

$y=A\cos {x}+B\sin {x}\,$,

where A, B are constants determined from boundary conditions. In the case where the equations are linear, this can be done by breaking the original equation down into smaller equations, solving those, and then adding the results back together. Unfortunately, many of the interesting differential equations are non-linear, which means that they cannot be broken down in this way. There are also a number of techniques for solving differential equations using a computer (see numerical ordinary differential equations).

Ordinary differential equations are to be distinguished from partial differential equations where $y$ is a function of several variables, and the differential equation involves partial derivatives.

Types of differential equations with some history

The influence of geometry, physics, and astronomy, starting with Newton and Leibniz, and further manifested through the Bernoullis, Riccati, and Clairaut, but chiefly through d'Alembert and Euler, has been very marked, and especially on the theory of linear partial differential equations with constant coefficients.

Homogeneous Linear ODEs with constant coefficients

The first method of integrating linear ordinary differential equations with constant coefficients is due to Euler, who made the solutions of

${\frac {d^{n}y}{dx^{n}}}+A_{1}{\frac {d^{n-1}y}{dx^{n-1}}}+\cdots +A_{n}y=0$

depend on those of the algebraic equation of the nth degree

$F(z)=z^{n}+A_{1}z^{n-1}+\cdots +A_{n}=0$

in which z^k takes the place of

${\frac {d^{k}y}{dx^{k}}}\quad \quad (k=1,2,\cdots ,n).$

This equation F(z) = 0, is the "characteristic" equation considered later by Monge and Cauchy.

If z is a (possibly not real) zero of F(z) of multiplicity m and $k\in \{0,1,\dots ,m-1\}\,$ then $y=x^{k}e^{zx}\,$ is a solution of the ODE. These functions make up a basis of the ODE's solutions.

If the A_i are real then real-valued solutions are preferable. Since the non-real z values will come in conjugate pairs, so will their corresponding ys; replace each pair with their linear combinations Re(y) and Im(y).

A case that involves complex roots can be solved with the aid of Euler's formula.

• Example: Given $y''-4y'+5=0\,$. The characteristic equation is $z^{2}-4z+5=0\,$ which has zeroes 2+i and 2−i. Thus the solution basis $\{y_{1},y_{2}\}$ is $\{e^{(2+i)x},e^{(2-i)x}\}\,$. Now y is a solution iff $y=c_{1}y_{1}+c_{2}y_{2}\,$ for $c_{1},c_{2}\in \mathbb {C}$.

Because the coefficients are real,

• we are likely not interested in the complex solutions
• our basis elements are mutual conjugates

The linear combinations

$u_{1}={\mbox{Re}}(y_{1})={\frac {y_{1}+y_{2}}{2}}=e^{2x}\cos(x)\,$ and

$u_{2}={\mbox{Im}}(y_{1})={\frac {y_{1}-y_{2}}{2i}}=e^{2x}\sin(x)\,$

will give us a real basis in $\{u_{1},u_{2}\}$.

Linear ODEs with constant coefficients

Suppose instead we face

${\frac {d^{n}y}{dx^{n}}}+A_{1}{\frac {d^{n-1}y}{dx^{n-1}}}+\cdots +A_{n}y=f(x)$

For later convenience, define the characteristic polynomial

$P(v)=v^{n}+A_{1}v^{n-1}+\cdots +A_{n}$

We find the solution basis $\{y_{1},y_{2},\ldots ,y_{n}\}$ as in the homogeneous (f=0) case. We now seek a particular solution y_p by the variation of parameters method. Let the coefficients of the linear combination be functions of x:

$y_{p}=u_{1}y_{1}+u_{2}y_{2}+\cdots +u_{n}y_{n}$

Using the "operator" notation $D=d/dx$ and a broad-minded use of notation, the ODE in question is $P(D)y=f$; so

$f=P(D)y_{p}=P(D)(u_{1}y_{1})+P(D)(u_{2}y_{2})+\cdots +P(D)(u_{n}y_{n})$

With the constraints

$0=u'_{1}y_{1}+u'_{2}y_{2}+\cdots +u'_{n}y_{n}$

$0=u'_{1}y'_{1}+u'_{2}y'_{2}+\cdots +u'_{n}y'_{n}$

…

$0=u'_{1}y_{1}^{(n-2)}+u'_{2}y_{2}^{(n-2)}+\cdots +u'_{n}y_{n}^{(n-2)}$

the parameters commute out, with a little "dirt":

$f=u_{1}P(D)y_{1}+u_{2}P(D)y_{2}+\cdots +u_{n}P(D)y_{n}+u'_{1}y_{1}^{(n-1)}+u'_{2}y_{2}^{(n-1)}+\cdots +u'_{n}y_{n}^{(n-1)}$

But $P(D)y_{j}=0$, therefore

$f=u'_{1}y_{1}^{(n-1)}+u'_{2}y_{2}^{(n-1)}+\cdots +u'_{n}y_{n}^{(n-1)}$

This, with the constraints, gives a linear system in the $u'_{j}$. This much can always be solved; in fact, combining Cramer's rule with the Wronskian,

$u'_{j}=(-1)^{n+j}f{\frac {W(y_{1},\ldots ,y_{j-1},y_{j+1}\ldots ,y_{n})}{W(y_{1},y_{2},\ldots ,y_{n})}}$

The rest is a matter of integrating $u'_{j}$.

The particular solution is not unique; $y_{p}+c_{1}y_{1}+\cdots +c_{n}y_{n}$ also satisfies the ODE for any set of constants c_j.

See also variation of parameters.

Example: Suppose $y''-4y'+5=sin(kx)$. We take the solution basis found above $\{e^{(2+i)x},e^{(2-i)x}\}$.

$W\,$	$={\begin{vmatrix}e^{(2+i)x}&e^{(2-i)x}\\(2+i)e^{(2+i)x}&(2-i)e^{(2-i)x}\end{vmatrix}}$
	$=e^{4x}{\begin{vmatrix}1&1\\2+i&2-i\end{vmatrix}}$
	$=-2ie^{4x}\,$

$u'_{1}\,$	$={\frac {1}{W}}{\begin{vmatrix}0&e^{(2-i)x}\\\sin(kx)&(2-i)e^{(2-i)x}\end{vmatrix}}$
	$=-{\frac {i}{2}}\sin(kx)e^{(-2-i)x}$

$u'_{2}\,$	$={\frac {1}{W}}{\begin{vmatrix}e^{(2+i)x}&0\\(2+i)e^{(2+i)x}&\sin(kx)\end{vmatrix}}$
	$={\frac {i}{2}}\sin(kx)e^{(-2+i)x}$

Using the list of integrals of exponential functions

$u_{1}\,$	$=-{\frac {i}{2}}\int \sin(kx)e^{(-2-i)x}\,dx$
	$={\frac {ie^{(-2-i)x}}{2(3+4i+k^{2})}}\left((2+i)\sin(kx)+k\cos(kx)\right)$

$u_{2}\,$	$={\frac {i}{2}}\int \sin(kx)e^{(-2+i)x}\,dx$
	$={\frac {ie^{(i-2)x}}{2(3-4i+k^{2})}}\left((i-2)\sin(kx)-k\cos(kx)\right)$

And so

$y_{p}\,$	$={\frac {i}{2(3+4i+k^{2})}}\left((2+i)\sin(kx)+k\cos(kx)\right)+{\frac {i}{2(3-4i+k^{2})}}\left((i-2)\sin(kx)-k\cos(kx)\right)$
	$={\frac {(5-k^{2})\sin(kx)+4k\cos(kx)}{(3+k^{2})^{2}+16}}$

(Notice that u₁ and u₂ had factors that canceled y₁ and y₂; that is typical.)

For interest's sake, this ODE has a physical interpretation as a driven damped harmonic oscillator; y_p represents the steady state, and $c_{1}y_{1}+c_{2}y_{2}$ is the transient.

Linear ODEs with variable coefficient

Method of undetermined coefficients

The method of undetermined coefficients (MoUC), is useful in finding solution for $y_{p}$. Given the ODE $P(D)y=f(x)$, find another differential operator $A(D)$ such that $A(D)f(x)=0$; this is called the annihilator. Applying $A(D)$ to both sides of the ODE gives an homogeneous ODE ${\big (}A(D)P(D){\big )}y=0$ for which we find a solution basis $\{y_{1},\ldots ,y_{n}\}$ as before. Then the original nonhomogeneous ODE is used to construct a system of equations restricting the coefficients of the linear combinations to satisfy the ODE.

Undetermined coefficients is not as general as variation of parameters in the sense that an annihilator does not always exist.

Example: Given $y''-4y'+5=\sin(kx)$, $P(D)=D^{2}-4D+5$. The simplest annihilator of $\sin(kx)$ is $A(D)=D^{2}+k^{2}$. The zeros of $A(z)P(z)$ are $\{2+i,2-i,ik,-ik\}$, so the solution basis of $A(D)P(D)$ is $\{y_{1},y_{2},y_{3},y_{4}\}=\{e^{(2+i)x},e^{(2-i)x},e^{ikx},e^{-ikx}\}$.

Setting $y=c_{1}y_{1}+c_{2}y_{2}+c_{3}y_{3}+c_{4}y_{4}$ we find

$\sin(kx)$	$=P(D)y$
	$=P(D)(c_{1}y_{1}+c_{2}y+c_{3}y_{3}+c_{4}y_{4})$
	$=c_{1}P(D)y_{1}+c_{2}P(D)y_{2}+c_{3}P(D)y_{3}+c_{4}P(D)y_{4}$
	$=0+0+c_{3}(-k^{2}-4ik+5)y_{3}+c_{4}(-k^{2}+4ik+5)y_{4}$
	$=c_{3}(-k^{2}-4ik+5)(\cos(kx)+i\sin(kx))+c_{4}(-k^{2}+4ik+5)(\cos(kx)-i\sin(kx))$

giving the system

$i=(k^{2}+4ik-5)c_{3}+(-k^{2}+4ik+5)c_{4}$

$0=(k^{2}+4ik-5)c_{3}+(k^{2}-4ik-5)c_{4}$

which has solutions

$c_{3}={\frac {i}{2(k^{2}+4ik-5)}}$, $c_{4}={\frac {i}{2(-k^{2}+4ik+5)}}$

giving the solution set

$y\,$	$=c_{1}y_{1}+c_{2}y_{2}+{\frac {i}{2(k^{2}+4ik-5)}}y_{3}+{\frac {i}{2(-k^{2}+4ik+5)}}y_{4}$
	$=c_{1}y_{1}+c_{2}y_{2}+{\frac {4k\cos(kx)-(k^{2}-5)\sin(kx)}{(k^{2}+4ik-5)(k^{2}-4ik-5)}}$
	$=c_{1}y_{1}+c_{2}y_{2}+{\frac {4k\cos(kx)+(5-k^{2})\sin(kx)}{k^{4}+6k^{2}+25}}$

Method of variation of parameters

As explained above, the general solution to a non-homogeneous, linear differential equation $y''(x)+p(x)y'(x)+q(x)y(x)=g(x)$ can be expressed as the sum of the general solution $y_{h}(x)$ to the corresponding homogenous, linear differential equation $y''(x)+p(x)y'(x)+q(x)y(x)=0$ and any one solution $y_{p}(x)$ to $y''(x)+p(x)y'(x)+q(x)y(x)=g(x)$.

Like the method of undetermined coefficients, described above, the method of variation of parameters is a method for finding one solution to $y''(x)+p(x)y'(x)+q(x)y(x)=g(x)$, having already found the general solution to $y''(x)+p(x)y'(x)+q(x)y(x)=0$. Unlike the method of undetermined coefficients, which fails except with certain specific forms of g(x), the method of variation of parameters will always work; however, it is significantly more difficult to use.

For a second-order equation, the method of variation of parameters makes use of the following fact:

Fact

Let p(x), q(x), and g(x) be functions, and let $y_{1}(x)$ and $y_{2}(x)$ be solutions to the homogeneous, linear differential equation $y''(x)+p(x)y'(x)+q(x)y(x)=0$. Further, let u(x) and v(x) be functions such that $u'(x)y_{1}(x)+v'(x)y_{2}(x)=0$ and $u'(x)y_{1}'(x)+v'(x)y_{2}'(x)=g(x)$ for all x, and define $y_{p}(x)=u(x)y_{1}(x)+v(x)y_{2}(x)$. Then $y_{p}(x)$ is a solution to the non-homogeneous, linear differential equation $y''(x)+p(x)y'(x)+q(x)y(x)=g(x)$.

Proof

$y_{p}(x)=u(x)y_{1}(x)+v(x)y_{2}(x)$

$y_{p}'(x)$	$=u'(x)y_{1}(x)+u(x)y_{1}'(x)+v'(x)y_{2}(x)+v(x)y_{2}'(x)$
	$=0+u(x)y_{1}'(x)+v(x)y_{2}'(x)$

$y_{p}''(x)$	$=u'(x)y_{1}'(x)+u(x)y_{1}''(x)+v'(x)y_{2}'(x)+v(x)y_{2}''(x)$
	$=g(x)+u(x)y_{1}''(x)+v(x)y_{2}''(x)$

$y_{p}''(x)+p(x)y'_{p}(x)+q(x)y_{p}(x)=g(x)+u(x)y_{1}''(x)+v(x)y_{2}''(x)+p(x)u(x)y_{1}'(x)+p(x)v(x)y_{2}'(x)+q(x)u(x)y_{1}(x)+q(x)v(x)y_{2}(x)$

$=g(x)+u(x)(y_{1}''(x)+p(x)y_{1}'(x)+q(x)y_{1}(x))+v(x)(y_{2}''(x)+p(x)y_{2}'(x)+q(x)y_{2}(x))=g(x)+0+0=g(x)$

Usage

To solve the second-order, non-homogeneous, linear differential equation $y''(x)+p(x)y'(x)+q(x)y(x)=g(x)$ using the method of variation of parameters, use the following steps:

1. Find the general solution to the corresponding homogeneous equation $y''(x)+p(x)y'(x)+q(x)y(x)=0$. Specifically, find two linearly independent solutions $y_{1}(x)$ and $y_{2}(x)$.
2. Since $y_{1}(x)$ and $y_{2}(x)$ are linearly independent solutions, their Wronskian $y_{1}(x)y_{2}'(x)-y_{1}'(x)y_{2}(x)$ is nonzero, so we can compute $-(g(x)y_{2}(x))/({y_{1}(x)y_{2}'(x)-y_{1}'(x)y_{2}(x)})$ and $({g(x)y_{1}(x)})/({y_{1}(x)y_{2}'(x)-y_{1}'(x)y_{2}(x)})$. If the former is equal to u'(x) and the latter to v'(x), then u and v satisfy the two constraints given above: that $u'(x)y_{1}(x)+v'(x)y_{2}(x)=0$ and that $u'(x)y_{1}'(x)+v'(x)y_{2}'(x)=g(x)$. We can tell this after multiplying by the denominator and comparing coefficients.
3. Integrate $-(g(x)y_{2}(x))/({y_{1}(x)y_{2}'(x)-y_{1}'(x)y_{2}(x)})$ and $({g(x)y_{1}(x)})/({y_{1}(x)y_{2}'(x)-y_{1}'(x)y_{2}(x)})$ to obtain u(x) and v(x), respectively. (Note that we only need one choice of u and v, so there is no need for constants of integration.)
4. Compute $y_{p}(x)=u(x)y_{1}(x)+v(x)y_{2}(x)$. The function $y_{p}$ is one solution of $y''(x)+p(x)y'(x)+q(x)y(x)=g(x)$.
5. The general solution is $c_{1}y_{1}(x)+c_{2}y_{2}(x)+y_{p}(x)$, where $c_{1}$ and $c_{2}$ are arbitrary constants.

Higher-order equations

The method of variation of parameters can also be used with higher-order equations. For example, if $y_{1}(x)$, $y_{2}(x)$, and $y_{3}(x)$ are linearly independent solutions to $y'''(x)+p(x)y''(x)+q(x)y'(x)+r(x)y(x)=0$, then there exist functions u(x), v(x), and w(x) such that $u'(x)y_{1}(x)+v'(x)y_{2}(x)+w'(x)y_{3}(x)=0$, $u'(x)y_{1}'(x)+v'(x)y_{2}'(x)+w'(x)y_{3}'(x)=0$, and $u'(x)y_{1}''(x)+v'(x)y_{2}''(x)+w'(x)y_{3}''(x)=g(x)$. Having found such functions (by solving algebraically for u'(x), v'(x), and w'(x), then integrating each), we have $y_{p}(x)=u(x)y_{1}(x)+v(x)y_{2}(x)+w(x)y_{3}(x)$, one solution to the equation $y'''(x)+p(x)y''(x)+q(x)y'(x)+r(x)y(x)=g(x)$.

Example

Solve the previous example, $y''+y=\sec x$ Recall $\sec x={\frac {1}{\cos x}}=f$. From technique learned from 3.1, LHS has root of $r=\pm i$ that yield $y_{c}=C_{1}\cos x+C_{2}\sin x$, (so $y_{1}=\cos x$, $y_{2}=\sin x$ ) and its derivatives

$\left\{{\begin{matrix}{{\dot {u}}={\frac {-y_{2}f}{W}}={\frac {-\sin x}{\cos x}}=\tan x}\\{{\dot {v}}={\frac {y_{1}f}{W}}={\frac {\cos x}{\cos x}}=1}\\\end{matrix}}\right.$

where the Wronskian

$W\left({y_{1},y_{2}:x}\right)=\left|{\begin{matrix}{\cos x}&{\sin x}\\{-\sin x}&{\cos x}\\\end{matrix}}\right|=1$

were computed in order to seek solution to its derivatives.

Upon integration,

$\left\{{\begin{matrix}u=-\int {\tan xdx=-\ln \left|{\sec x}\right|+C}\\v=\int {1dx=x+C}\\\end{matrix}}\right.$

Computing $y_{p}$ and $y_{G}$:

${\begin{matrix}y_{p}=f=uy_{1}+vy_{2}=\cos x\ln \left|{\cos x}\right|+x\sin x\\y_{G}=y_{c}+y_{p}=C_{1}\cos x+C_{2}\sin x+x\sin x+\cos x\ln \left({\cos x}\right)\\\end{matrix}}$

General solution method for first-order linear ODEs

For a first-order linear ODE, with coefficients that may or may not vary with t:

$x'(t)+p(t)x(t)=r(t)$

Then,

$x=e^{-a(t)}\left(\int r(t)e^{a(t)}dt+\kappa \right)$

where $\kappa$ is the constant of integration, and

$a(t)=\int {p(s)ds}.$

Proof

This proof comes from Jean Bernoulli. Let

$x^{\prime }+px=r$

Suppose for some unknown functions u(t) and v(t) that x = uv.

Then

$x^{\prime }=u^{\prime }v+uv^{\prime }$

Substituting into the differential equation,

$u^{\prime }v+uv^{\prime }+puv=r$

Now, the most important step: Since the differential equation is linear we can split this into two independent equations and write

$u^{\prime }v+puv=0$

$uv^{\prime }=r$

Since v is not zero, the top equation becomes

$u^{\prime }+pu=0$

The solution of this is

$u=e^{-\int pdt}$

Substituting into the second equation

$v=\int re^{\int pdt}+C$

Since x = uv, for arbitrary constant C

$x=e^{-\int pdt}\left(\int re^{\int pdt}+C\right)$

First order differential equation with constant coefficients

As an illustrative example, consider a first order differential equation with constant coefficients:

$a{\frac {dx}{dt}}+bx=Af(t).$

This equation is particularly relevant to first order systems such as RC circuits, mass-damper systems.

After nondimensionalization, the equation becomes

${\frac {d\chi }{d\tau }}+\chi =F(\tau ).$

In this case, p(t) = r(t) = 1.

Hence its solution by inspection is

$\chi (\tau )=e^{-\tau }\left(\int F(\tau )e^{\tau }\,d\tau +C\right).$

Linear PDEs

The theory of linear partial differential equations may be said to begin with Lagrange (1779 to 1785). Monge (1809) treated ordinary and partial differential equations of the first and second order, uniting the theory to geometry, and introducing the notion of the "characteristic", the curve represented by $F(z)=0$, which was investigated by Darboux, Levy, and Lie.

First-order PDEs

Pfaff (1814, 1815) gave the first general method of integrating partial differential equations of the first order, of which Gauss (1815) gave an analysis. Cauchy (1819) gave a simpler method, attacking the subject from the analytical standpoint, but using the Monge characteristic. Cauchy also first stated the theorem (now called the Cauchy-Kovalevskaya theorem) that every analytic differential equation defines an analytic function, expressible by means of a convergent series.

Jacobi (1827) also gave an analysis of Pfaff's method, besides developing an original one (1836) which Clebsch published (1862). Clebsch's own method appeared in 1866, and others are due to Boole (1859), Korkine (1869), and A. Mayer (1872). Pfaff's problem (on total differential equations) was investigated by Natani (1859), Clebsch (1861, 1862), DuBois-Reymond (1869), Cayley, Baltzer, Frobenius, Morera, Darboux, and Lie.

The next great improvement in the theory of partial differential equations of the first order was made by Lie (1872), who placed the whole subject on a solid foundation. After about 1870, Darboux, Kovalevsky, Méray, Mansion, Graindorge, and Imschenetsky became prominent in this line. The theory of partial differential equations of the second and higher orders, beginning with Laplace and Monge, was notably advanced by Ampère (1840).

The integration of partial differential equations with three or more variables was the object of elaborate investigations by Lagrange, and his name became connected with certain subsidiary equations. It was he and Charpit who originated one of the methods for integrating the general equation with two variables; a method which now bears Charpit's name.

Singular solutions

The theory of singular solutions of ordinary and partial differential equations was a subject of research from the time of Leibniz, but only since the middle of the nineteenth century did it receive special attention. A valuable but little-known work on the subject is that of Houtain (1854). Darboux (starting in 1873) was a leader in the theory, and in the geometric interpretation of these solutions he opened a field which was worked by various writers, notably Casorati and Cayley. To the latter is due (1872) the theory of singular solutions of differential equations of the first order as accepted circa 1900.

Reduction to quadratures

The primitive attempt in dealing with differential equations had in view a reduction to quadratures. As it had been the hope of eighteenth-century algebraists to find a method for solving the general equation of the $n$th degree, so it was the hope of analysts to find a general method for integrating any differential equation. Gauss (1799) showed, however, that the differential equation meets its limitations very soon unless complex numbers are introduced. Hence analysts began to substitute the study of functions, thus opening a new and fertile field. Cauchy was the first to appreciate the importance of this view. Thereafter the real question was to be, not whether a solution is possible by means of known functions or their integrals, but whether a given differential equation suffices for the definition of a function of the independent variable or variables, and if so, what are the characteristic properties of this function.

The Fuchsian theory

Two memoirs by Fuchs (Crelle, 1866, 1868), inspired a novel approach, subsequently elaborated by Thomé and Frobenius. Collet was a prominent contributor beginning in 1869, although his method for integrating a non-linear system was communicated to Bertrand in 1868. Clebsch (1873) attacked the theory along lines parallel to those followed in his theory of Abelian integrals. As the latter can be classified according to the properties of the fundamental curve which remains unchanged under a rational transformation, so Clebsch proposed to classify the transcendent functions defined by the differential equations according to the invariant properties of the corresponding surfaces f = 0 under rational one-to-one transformations.

Lie's theory

From 1870 Lie's work put the theory of differential equations on a more satisfactory foundation. He showed that the integration theories of the older mathematicians can, by the introduction of what are now called Lie groups, be referred to a common source; and that ordinary differential equations which admit the same infinitesimal transformations present comparable difficulties of integration. He also emphasized the subject of transformations of contact (Berührungstransformationen).

See also

• Examples of differential equations
• Differential equations of mathematical physics
• Differential equations from outside physics
• Difference equation
• Laplace transform applied to differential equations
• Boundary value problem
• List of dynamical systems and differential equations topics

External links

• EqWorld: The World of Mathematical Equations, containing a list of ordinary differential equations with their solutions.
• Example ODEs from exampleproblems.com.

Bibliography

• A. D. Polyanin and V. F. Zaitsev, Handbook of Exact Solutions for Ordinary Differential Equations (2nd edition)", Chapman & Hall/CRC Press, Boca Raton, 2003. ISBN 1-58488-297-2
• A. D. Polyanin, V. F. Zaitsev, and A. Moussiaux, Handbook of First Order Partial Differential Equations, Taylor & Francis, London, 2002. ISBN 0-415-27267-X
• D. Zwillinger, Handbook of Differential Equations (3rd edition), Academic Press, Boston, 1997.
• Refaat El Attar, Ordinary Differential Equations, Lulu Press, Morrisville NC, 2005. ISBN 1-41163-920-0.
• Hartman, Philip, Ordinary Differential Equations, 2nd Ed., Society for Industrial & Applied Math, 2002. ISBN 0898715105.